Integration is the operation which is inverse to differentiation. This operation is very important for theoretical physics, and you should become familiar with computing integrals.

Definite integrals are written as follows: . (There is really no separate concept of an "indefinite integral." The notation is merely the shorthand for , where is not used in later calculations and/or is chosen for convenience in some natural way.)

Some basic integrals:

Substitution of variable: introduce , where is a function:

Examples:

Integration by parts:

For example (here we do not write the limits of integration):

Note: trigonometric functions are sometimes more conveniently written in terms of complex exponentials (using the Euler formula):

In this section, differential equations are written for unknown function . Derivatives are denoted by overdots: , , etc.

The general solution of a differential equation is a function that solves the equation and contains arbitrary constants. For equations with first derivatives (first-order equations) there is only one constant; for second-order equations there are two constants, etc.

Examples:

Find the general solution of the equation . Answer: ,

where is an arbitrary constant.

Find the general solution of . Answer: .

Find the general solution of . Answer: .

Find the general solution of . Answer: .

Find the general solution of the second-order equation . Answer: .

Find the general solution of . Answer: .

Find the general solution of . Answer: .

Find the general solution of . Solution: we look for . Then must be such that .

This has two solutions, and . So the general solution is .

Exercises:

Find the general solution of the following equations.

Particular solutions are selected from general solutions by conditions, such as the initial conditions:

The general solution of is . The conditions , are satisfied only if , . Therefore, the particular solution of equation (1) is .

To find particular solutions, we first find the general solution with arbitrary constants and then determine the values of these constants using the initial conditions.

Exercises:

Solve the following equations with initial conditions. Plot the resulting functions .

Boundary-value problems are differential equations with conditions at different points. Note: There are usually infinitely many functions that solve a differential equation. The general solution represents all these functions by means of a formula with arbitrary constants. A particular solution is selected by conditions, and one needs as many conditions as unknown constants. So, e.g. for a second-order differential equation, there are two arbitrary constants, and one needs two conditions to specify a unique solution. Instead of specifying two conditions conditions at the same time, such as , one can specify two conditions at different times, e.g. . Such conditions are called boundary conditions.

Equations of the form have the general solution . What about ? The general solution is . How to guess? Write and substitute; then find the correct value of .

Exercises:

Solve the following equations.

Hint: In the last equation, replace by an unknown function .

Similarly, have the general solution . What about ? Write and find the correct values of and , then add the general solution. The result is .

Note that in every case the general solution of the equation without the right-hand side (the homogeneous equation) is added to a guessed solution of the equation with the right-hand side (the inhomogeneous equation). This is the general principle when dealing with such equations.

For example, the differential equations and are of this form. To solve these equations, we use the trick called "separation of variables." We look for the solution of the form , where and are some functions. If the solution were in this form, then , which is the same as . This should be equivalent to the

original differential equation . Therefore,

These equations are easy to solve:

We can compute these functions and find the general solution of the original

equation in the following implicit form:

Here, and are arbitrary constants of integration. This solution satisfies the initial condition .

Example:

Consider the equation . We write

Note that only one constant of integration is necessary, despite the presence of two "indefinite integrals".

One case is second-order equations with "source" (i.e. with nonzero function in the right hand side). We need to guess a solution of the inhomogeneous equation. We guess the solution by writing an ansatz with unknown coefficients. Here are some examples:

Note: in the last example, we need a term because and are already solutions of the homogeneous equation!

Another example:

We look for solutions in the form , find and only one root . Then we use a special trick: the general solution is not but .

Exercises:

Solve the following equations.

Systems of differential equations: for example,

may be solved either by differentiation: , or by guessing the solution in the form , .

Note: Since these equations are linear, you should add all the possible pieces of the general solution with different values of .

Exercise:

By guessing the solution in the form , find the general solution of the system